Guidance Concept for a Mars Ascent Vehicle First Stage

Transcription

1 NASA/TM Guidance Concept for a Mars Ascent Vehicle First Stage Eric M. Queen Langley Research Center, Hampton, Virginia November 2000

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3 NASA/TM Guidance Concept for a Mars Ascent Vehicle First Stage Eric M. Queen Langley Research Center, Hampton, Virginia National Aeronautics and Space Administration Langley Research Center Hampton, Virginia November 2000

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5 Guidance Concept for a Mars Ascent Vehicle First Stage Eric M. Queen NASA Langley Research Center Hampton, VA Abstract This paper presents a guidance concept for use on the rst stage of a candidate Mars Ascent Vehicle (MAV). The guidance is based on a calculus of variations approach similar to that used for the nal phase of the Apollo Earth return guidance. A three degree-offreedom (3DOF) Monte Carlo simulation is used to evaluate performance and robustness of the algorithm. 1 Nomenclature E Energy (m 2 =s 2 ) g Gravitational acceleration (m=s 2 ) H Hamiltonian H Angular Momentum (m 2 =s) h Altitude (m) L Lift (N) m Vehicle mass (kg) R e Planetary radius (m) r Radius from planet center (m) T Thrust (N) t Time (s) V Velocity (m/s) Angle of attack (Thrust Angle) (deg) Out-of-plane thrust angle (deg) Flight path angle (deg) Partial Derivative Costate Gravitational parameter (m 3 =s 2 ) Latitude (deg) BO a targ Dummy variable for time Cost functional Heading angle (deg) Value at Burnout Value at Apoapsis Target Value 2 Introduction Although eorts are underway to continue improvements in reliability and sensitivity of robotic planetary probes, they will not, in the forseeable future, be able to match the examination and analysis capabilities available here on Earth. One solution to this dilemna is to retrieve planetary samples for analysis here. This has been proposed for samples from Mars starting with the 2003 launch opportunity. One plan calls for a lander to be sent to Mars to collect soil samples and launch them into orbit around Mars. The samples will remain in orbit until the orbiter and lander launched in 2005 reach Mars. The 2005 lander will collect more samples and launch them into Mars orbit, and the 2005 orbiter will then rendezvous with both sets of samples and return them to Earth. The Mars Ascent Vehicle (MAV) is subject to severe design constraints. In addition to the usual premiums on weight, volume, and budget, the MAV must oper- 1

6 346.4 mm mm mm Reorient, spin-up, stage 2 burn 200 sec coast 20 sec boost Figure 1. MAV Conguration. ate somewhat autonomously after being subjected, unattended, to a severe environment for nearly a year. As a result of these and other constraints, The MAV has unique challenges in its design, especially for guidance and control. Because of the long travel times, the MAV will have solid-fuel motors. Figure 1 shows the general conguration considered in this report. The ascent trajectory from the Martian surface begins with a high thrust phase that lasts approximately 20 seconds. The MAV then coasts for approximately 200 seconds, at which time it repoints, spins up to 20 rpm, separates the spent rst stage and res the second stage. One half orbit later, the third stage motor (which is mounted backwards to the other stages) is red to circularize the - nal orbit. The second and third stages are not guided, though the repointing maneuver may be modied to account for an o-nominal rst stage burn/coast. This sequence is illustrated in gure 2. While the rst stage motor is burning, the vehicle is controlled byvanes in the rocket exhaust. After rst stage burnout, aerodynamic surfaces are available to reorient the vehicle, but because of the low density of the Martian atmosphere, the ability to adjust the rst stage trajectory is limited. Thus, the objective of the rst stage guidance is to achieve the highest degree of accuracy in the desired burnout conditions, subject to uncertainties in the winds, atmospheric density, vehicle/payload mass, and total impulse of the motor. The short burn time requires that the Mars Stage 3 circularization burn 1/2 orbit coast Figure 2. MAV Ascent Sequence. rst stage guidance be very fast and robust to a rapidly changing plant. The scheme employed for guidance during the rst stage uses an approach similar to that used for the nal phase of the Apollo Entry Guidance [1, 2, 3]. Based on a nominal trajectory, the sensitivities of the nal state (here the burnout state) to variations in the current state are determined and used to drive those variations to zero at the nal time. The next section gives the derivation of the feedback equations for both the in-plane and out-of-plane components. Section 4 describes the implementation of the algorithm in a numerical simulation and describes some results of that implementation. 3 Theoretical Development The in-plane dierential dynamical equations for a rocket ascent are 2

7 as follows. For altitude, _h = V sin() (1) where h is the altitude, V is the velocity, and is the ight path angle. The equation for velocity is _V = T cos() m, g sin() (2) and the equation for ight path angle is _ = T sin() mv + V 2 R e + h, g! cos() (3) The function to be maximized is the energy at burnout, so the Hamiltonian is [4]: T cos() H = h V sin()+ V m T sin(), V g sin()+ mv V 2 cos() + (R e + h), g cos() and the costate equations are: _ = V 2 cos() (R e + h) 2 _ =, h sin(), 2 V cos() R e + h + T sin() mv 2 =, h V cos()+g V cos() + V 2 sin(), g sin() R e + h (4) (5) (6) (7) From [1], assuming that the perturbation in the control will be constant, where or =,T (t)x(t) (t) (t) =,Z t t f _ (t) = V T sin() m (8) T d T cos() mv (10) The above equations (5), (6), (7), (10) are integrated backwards from the nal condition using states from the nominal trajectory. It is desired to match apoapsis to put second stage burn at the right position. So, let =,(r a, r targ ) 2 (11) The boundary conditions for the costates are: (t h (t V (t From orbital mechanics: = 2(r a, r targ (12) = 2(r a, r targ (13) = 2(r a, r targ (14) V BO r BO cos( BO )=V a r a (15) and V 2 BO 2, = V 2 a r BO 2, (16) r a where the subscript 0 denotes quantities at burnout and the subscript a denotes quantities at apoapsis. Dene the energy and angular momentum E = V 2 BO 2, r BO (17) 3

8 Solving, H = V BO r BO cos( BO =,r BOV BO sin BO (30) r a = r BOV BO cos BO V a = H V a (19) substitute eqn(19) into eqn(16) and solve for V a : V a = p 2 +2EH 2 (20) H The higher velocity is the periapsis root. The minus root is desired. Note: Energy will be negative for an elliptic orbit. The velocity at apoapsis will still be positive. V a =, p 2 +2EH 2 (21) H Substituting into eqn(19) and simplifying r a = H 2, p 2 +2EH 2 (22) The boundary conditions on the costates are = V a V = V BO cos = r 2 (25) BO The V BO and BO derivatives are exactly analogous, = V a V = r BO cos = V BO = V a V =0 (31) Note that the nal state will never depend on the nal control so the \control costate",, will always have a nal condition of 0. When this is used in equation (8) it implies that a state perturbation at the nal time requires innite control to be corrected at the nal time, i.e. instantaneously. Also note that, for ight implementation, very little of this process occurs onboard. The costates,, V, etc, are determined from a nominal trajectory prior to ight. The costates are stored as tables or polynomials and the control is then a simple function of these stored values and the current state. The out-of-plane equation is _ = L sin()+t sin() mv cos(), V r cos() cos() tan() (32) where is the latitude, is the out-of-plane thrust angle (similar to sideslip angle), and L is the lift force. It is assumed that the lift force is negligible compared to the T sin() term and that the entire ight takes place near enough to the equator that the last term can be neglected. Writing _ as a nite dierence and solving for sin(), results in new, old! mv cos() sin() = (33) t T where new is the commanded heading angle and old is the current heading angle. For this implementation, the command was chosen as a ramp in time that brings the nominal trajectory to a 45 degree inclination at burnout. 4

9 4 Numerical Simulation Results The guidance algorithm described above was implemented in a numerical 3DOF simulation using the Program to Optimize Simulated Trajectories (POST) program [5]. The simulation was incorporated into a Monte Carlo analysis with dispersions as listed in Table 1. Table 1. Monte Carlo Inputs Variable Range Dist Launch Altitude 0-2 km U Launch Latitude 0.1 deg G Launch Longitude 0.1 deg G Launch Azimuth deg G Launch FPA deg G E-W Wind 50 m/s U N-S Wind 5-30 m/s U Propellant Mass kg 0.3% G Payload Mass kg G Thrust N 4.0% G I sp 279s 1% G C A 5% G C N 5% G The rst column of Table 1 lists the quantities that were dispersed within the limits shown in the second column. The nal column denotes the type of random distribution sampled; 'G' for Gaussian and 'U' for uniform. Random atmosphere variations were also included based on MarsGRAM [6]. The simulation was executed 2000 times with these dispersions. For this mission, because of the long (uncontrolled) coast phase and the need for eventual rendezvous by the orbiter, the apoapsis and inclination are the most critical parameters. Figure 3 shows the nal apoapsis and inclination for 2000 cases. For all cases the apoapsis is between 97 and 103 km, and the inclination is between 44.7 and 45.3 degrees. Table 2 summarizes some statistics from the Monte Carlo simulation. Apoapsis (km) Inclination (deg) Figure 3. Burnout Conditions for 2000 cases. Table 2. Monte Carlo Statistics Variable Mean Max Min Altitude km Inclination deg Apoapsis km Periapsis km Tot. Impulse kn*s 5 Conclusions A guidance algorithm for the rst stage of a proposed Mars Ascent Vehicle has been developed. This algorithm is based on a calculus of variations approach, using inuence coecients to drive the vehicle state to a desired terminal state. The algorithm is designed to provide good performance with very little on-board computation. While the exact conguration is subject to change, this algorithm is potentially useful across a wide range of applications. The proposed guidance algorithm has 5

10 been implemented and tested in a 3DOF Monte Carlo simulation. The results show that the algorithm controls the vehicle to relatively tight tolerances under reasonable environmental dispersions, keeping the nal condition within about a quarter degree in inclination and three kilometers apoapsis. References [1] Ro, Ted and Queen, Eric, \Study of Martian Aerocapture Terminal Point Guidance," AIAA , AIAA Atmospheric Flight Mechanics Conference, August 10-12, 1998, Boston, Ma. [2] Carman, G., Ives, D., and Geller, D., \Apollo-Derived Mars Precision Lander Guidance," AIAA ,AIAA Atmospheric Flight Mechanics Conference, August 10-12, 1998, Boston, Ma. [3] Guidance and Navigation for Entry Vehicles, NASA SP-8015, Nov [4] Bryson, A. E., and Ho, Y.-C., Applied Optimal Control, Hemisphere Publishing Corp., [5] Bauer, G.L., Cornick, D.E., and Stevenson, T. \Capabilities and Applications of the Program to Optimize Simulated Trajectories (POST)," NASA CR-2770, February, [6] Justus,C.G., \Mars Global Reference Atmospheric Model for Mission Planning and Analysis," Journal of Spacecraft and Rockets,Vol.28, No.2, pp , March-April,

11 REPORT DOCUMENTATION PAGE Form Approved OMB No Public reporting burden for this collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing this burden, to Washington Headquarters Services, Directorate for Information Operations and Reports, 1215 Jefferson Davis Highway, Suite 1204, Arlington, VA , and to the Office of Management and Budget, Paperwork Reduction Project ( ), Washington, DC AGENCY USE ONLY (Leave blank) 2. REPORT DATE November TITLE AND SUBTITLE Guidance Concept for a Mars Ascent Vehicle First Stage 3. REPORT TYPE AND DATES COVERED Technical Memorandum 5. FUNDING NUMBERS WU AUTHOR(S) Eric M. Queen 7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) NASA Langley Research Center Hampton, VA PERFORMING ORGANIZATION REPORT NUMBER L SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES) National Aeronautics and Space Administration Washington, DC SPONSORING/MONITORING AGENCY REPORT NUMBER NASA/TM SUPPLEMENTARY NOTES 12a. DISTRIBUTION/AVAILABILITY STATEMENT 12b. DISTRIBUTION CODE Unclassified-Unlimited Subject Category 15 Distribution: Nonstandard Availability: NASA CASI (301) ABSTRACT (Maximum 200 words) This paper presents a guidance concept for use on the first stage of a Mars Ascent Vehicle (MAV). The guidance is based on a calculus of variations approach similar to that used for the final phase of the Apollo Earth return guidance. 14. SUBJECT TERMS Guidance algorithm; Mars Ascent Vehicle 17. SEC U RITY CL ASSIF IC AT ION O F REPO R T Unclassified 18. SEC U RITY CL ASSIF IC AT ION O F TH IS PA GE Unclassified 19. SECURITY CLASSIFICATION OF ABSTRACT Unclassified 15. NUMBER OF PAGES PRICE CODE A LIMITATION OF ABSTRACT UL NSN Standard Form 298 (Rev. 2-89) Prescribed by ANSI Std. Z